This exercise is motivated by recent discussions about the filibuster in the U$.$S$.$ Senate. The filibuster is a rule in the U$.$S$.$ Senate that effectively requires a supermajority of senators to approve a policy change. Since that means a larger number of veto players, the rule is blamed for a lot of policy inaction in the U$.$S$.$ Congress. In particular, since typically no party has a required supermajority, the filibuster prevents the party controlling the majority of the Senate from pursuing its policy agenda.

An interesting fact about the filibuster is that it could be abandoned using simple majority of votes. Hence, if a simple majority of legislators are frustrated by their inability to pursue a policy change, they can remove the filibuster first, and then pursue the desired changes. Many commentators are puzzled why this has not happened yet. The exercise below provides one possible explanation.

Consider a legislature consisting of five legislators $($A$,$ B$,$ C$,$ D$,$ E$)$ with ideal points: x$$A $=1,$ x$$B $=2,$ x$$C $=3,$ x$$D $=4,$ x$$E $=5.$ The legislators have symmetric, single$-$peaked preferences $($i$.$e$.$ they always prefer policies closer to their ideal point$).$

The status quo is xsq $=1.$ Legislator E is the leader of the chamber $($agenda setter$)$ who can propose a new policy, x$.$ After the leader proposes a policy, the legislature can accept or reject the proposal. If the proposed policy is rejected, the status quo remains in place. If the proposed policy is accepted, the new policy is implemented. All legislators $($including the leader$)$ vote on the proposal.

$($a$)$For legislators A$,$ B$,$ C and D$,$ find out the set of policies that each of them is willing to accept relative to the status quo.

$($b$)$Suppose the legislature uses a simple majority rule, i$.$e$.$ a proposal is accepted if at least $3$ legislators accepts it$.$ What policy will legislator E propose in equilibrium? $($Hint: Legislator E needs at least $2$ additional votes from legislators A$,$ B$,$ C and D$.$ Which $2$ legislators are willing to accept policies that is the closest to E$$s ideal point? Those will be the veto players that E will try to convince to accept E$$s proposal.$)$

$($c$)$Suppose the legislature uses a supermajority rule, i$.$e$.$ a proposal is accepted if at least $4$ legislators accepts it$.$ What policy will legislator E propose in equilibrium? $($Hint: Legislator E needs at least $3$ additional votes from legislators A$,$ B$,$ C and D$.$ Which $3$ legislators has are willing to accept policies that is the closest to E$$s ideal point?Those will be the veto players that E will try to convince to accept E$$s proposal.$)$

$($d$)$Is the median legislator, i$.$e$.$ legislator C$,$ better off or worse off under a supermajority rule?

$($e$)$Use the insights you obtain from previous questions, briefly explain why political parties do not often attempt to remove the filibuster despite holding a simple majority.